Signal Processing First Lecture 10 Filtering Intro 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer READING ASSIGNMENTS This Lecture: Chapter 5, Sects. 5-1, 5-2 and 5-3 (partial) Other Reading: Recitation: Ch. 5, Sects 5-4, 5-6, 5-7 and 5-8 CONVOLUTION Next Lecture: Ch 5, Sects. 5-3, 5-5 and 5-6 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer LECTURE OBJECTIVES INTRODUCE FILTERING IDEA Weighted Average Running Average FINITE IMPULSE RESPONSE FILTERS FIR Filters Show how to compute the output y[n] from the input signal, x[n] 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer DIGITAL FILTERING x(t) A-to-D x[n] COMPUTER y[n] D-to-A y(t) CONCENTRATE on the COMPUTER PROCESSING ALGORITHMS SOFTWARE (MATLAB) HARDWARE: DSP chips, VLSI DSP: DIGITAL SIGNAL PROCESSING 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer The TMS32010, 1983 First PC plug-in board from Atlanta Signal Processors Inc. 5/27/2018 © 2003, JH McClellan & RW Schafer

Rockland Digital Filter, 1971 For the price of a small house, you could have one of these. 5/27/2018 © 2003, JH McClellan & RW Schafer

Digital Cell Phone (ca. 2000) 5/27/2018 © 2003, JH McClellan & RW Schafer Now it plays video

© 2003, JH McClellan & RW Schafer DISCRETE-TIME SYSTEM OPERATE on x[n] to get y[n] WANT a GENERAL CLASS of SYSTEMS ANALYZE the SYSTEM TOOLS: TIME-DOMAIN & FREQUENCY-DOMAIN SYNTHESIZE the SYSTEM x[n] COMPUTER y[n] 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer D-T SYSTEM EXAMPLES EXAMPLES: POINTWISE OPERATORS SQUARING: y[n] = (x[n])2 RUNNING AVERAGE RULE: “the output at time n is the average of three consecutive input values” x[n] SYSTEM y[n] 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer DISCRETE-TIME SIGNAL x[n] is a LIST of NUMBERS INDEXED by “n” STEM PLOT 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer 3-PT AVERAGE SYSTEM ADD 3 CONSECUTIVE NUMBERS Do this for each “n” Make a TABLE n=0 5/27/2018 © 2003, JH McClellan & RW Schafer n=1

© 2003, JH McClellan & RW Schafer INPUT SIGNAL OUTPUT SIGNAL 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer PAST, PRESENT, FUTURE “n” is TIME 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer ANOTHER 3-pt AVERAGER Uses “PAST” VALUES of x[n] IMPORTANT IF “n” represents REAL TIME WHEN x[n] & y[n] ARE STREAMS 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer GENERAL FIR FILTER FILTER COEFFICIENTS {bk} DEFINE THE FILTER For example, 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer GENERAL FIR FILTER FILTER COEFFICIENTS {bk} FILTER ORDER is M FILTER LENGTH is L = M+1 NUMBER of FILTER COEFFS is L 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer GENERAL FIR FILTER SLIDE a WINDOW across x[n] x[n-M] x[n] 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer FILTERED STOCK SIGNAL INPUT OUTPUT 5/27/2018 © 2003, JH McClellan & RW Schafer 50-pt Averager

© 2003, JH McClellan & RW Schafer SPECIAL INPUT SIGNALS x[n] = SINUSOID x[n] has only one NON-ZERO VALUE FREQUENCY RESPONSE (LATER) UNIT-IMPULSE 1 n 5/27/2018 © 2003, JH McClellan & RW Schafer

UNIT IMPULSE SIGNAL d[n] d[n] is NON-ZERO When its argument is equal to ZERO 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer MATH FORMULA for x[n] Use SHIFTED IMPULSES to write x[n] 5/27/2018 © 2003, JH McClellan & RW Schafer

SUM of SHIFTED IMPULSES This formula ALWAYS works 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer 4-pt AVERAGER CAUSAL SYSTEM: USE PAST VALUES INPUT = UNIT IMPULSE SIGNAL = d[n] OUTPUT is called “IMPULSE RESPONSE” 5/27/2018 © 2003, JH McClellan & RW Schafer

4-pt Avg Impulse Response d[n] “READS OUT” the FILTER COEFFICIENTS n=0 “h” in h[n] denotes Impulse Response NON-ZERO When window overlaps d[n] 1 n n=–1 n=0 n=1 n=4 n=5 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer FIR IMPULSE RESPONSE Convolution = Filter Definition Filter Coeffs = Impulse Response CONVOLUTION 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer FILTERING EXAMPLE 7-point AVERAGER Smooth compared to 3-point Averager By making its amplitude (A) smaller 3-point AVERAGER Changes A slightly 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer 3-pt AVG EXAMPLE USE PAST VALUES 5/27/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer 7-pt FIR EXAMPLE (AVG) CAUSAL: Use Previous 5/27/2018 © 2003, JH McClellan & RW Schafer LONGER OUTPUT